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Hilbert-Geo: Solving Solid Geometric Problems by Neural-Symbolic Reasoning

CVPR 2026 · Vision-language and multimodal › Multimodal LLMs · approach: Spatial reasoning in VLMs · 0 citations (OpenAlex)

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Introduces Hilbert-Geo, a neural-symbolic framework that parses solid and plane geometry problems into a formal CDL language with an MLLM and solves them with a symbolic reasoning engine, reaching 77.3% on the new SolidFGeo2k benchmark.

Problem

MLLMs perform poorly on solid (3D) geometry problems because of 3D spatial diagram understanding, visual perception errors, and complex multi-step reasoning, and existing solid geometry benchmarks are flawed or contaminated.

Approach

A Parse2Reason pipeline: an MLLM few-shot parses text and diagram into a formal conditional description language (CDL) built from a solid-geometry predicate library, then a symbolic search engine applies a 220-theorem bank to derive verifiable solutions.

Key results

• Hilbert-Geo (Gemini 2.5 Pro parser) reaches 77.3% on SolidFGeo2k versus 54.2% for Gemini 2.5 Pro alone.
• Hilbert-Geo reports 84.1% on MathVerse-Solid versus 62.9% for GPT-5 and 80.2% on PlaneFGeo3k.
• With 15 few-shot samples, the pipeline reaches 42.1% on SolidFGeo2k, above DeepSeek-V3's peak.

Surprising findings

• Hilbert-Geo with ground-truth CDL reaches only 78.7%, so the symbolic engine itself caps accuracy below 80%.
• Hilbert-Geo's reported 77.3% and Gemini 2.5 Pro's 54.2% are overall accuracies; the MathVerse-Solid figure is reported only in the abstract and text.

Limitations

• Symbolic engine accuracy with ground-truth CDL is only 78.7%, and combinatorial explosion causes timeouts on some problems.
• Parsing quality depends on the upstream MLLM; errors in CDL distortion, hallucinations, and visual perception remain.
• Reported text contains inconsistencies (e.g., 'Hilbert' vs Hilbert-Geo, mismatched subject-percentage claims, and an inconsistent 'no reasoning errors' claim).

Experimental setup

Datasets: SolidFGeo2k PlaneFGeo3k MathVerse-Solid SolidGeo

Benchmarks: SolidFGeo2k PlaneFGeo3k MathVerse-Solid

Metrics: accuracy fuzzy Jaccard similarity average time reasoning steps

Baselines: GPT-4o GPT-5 Claude 3.7 Sonnet Gemini 2.5 Flash Gemini 2.5 Pro Llama 3.3 70B Qwen2.5-VL-Instruct-32B Qwen2.5-VL-Instruct-7B DeepSeek-V3

Builds on: FormalGeo

Closest papers by meaning

• Geoint-R1: Formalizing Multimodal Geometric Reasoning with Dynamic Auxiliary Constructions (CVPR 2026 · similarity 0.92)
• SolidGeo: Measuring Multimodal Spatial Math Reasoning in Solid Geometry (NeurIPS 2025 · similarity 0.91)
• GeoX: Geometric Problem Solving Through Unified Formalized Vision-Language Pre-training (ICLR 2025 · similarity 0.89)
• GeoBench: Rethinking Multimodal Geometric Problem-Solving via Hierarchical Evaluation (ICLR 2026 · similarity 0.89)
• GNS: Solving Plane Geometry Problems by Neural-Symbolic Reasoning with Multi-Modal LLMs (AAAI 2025 · similarity 0.88)
• GeoTikzBridge: Advancing Multimodal Code Generation for Geometric Perception and Reasoning (CVPR 2026 · similarity 0.87)